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Massachusetts Institute of Technology

I NTRODUCTION

to

A PPLIED N UCLEAR P HYSICS

Spring 2012

Prof. Paola Cappellaro

Nuclear Science and Engineering Department

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• 1 Introduction to Nuclear Physics
  • 1.1 Basic Concepts
    • 1.1.1 Terminology
    • 1.1.2 Units, dimensions and physical constants
    • 1.1.3 Nuclear Radius
  • 1.2 Binding energy and Semi-empirical mass formula
    • 1.2.1 Binding energy
    • 1.2.2 Semi-empirical mass formula
    • 1.2.3 Line of Stability in the Chart of nuclides
  • 1.3 Radioactive decay
    • 1.3.1 Alpha decay
    • 1.3.2 Beta decay
    • 1.3.3 Gamma decay
    • 1.3.4 Spontaneous fission
    • 1.3.5 Branching Ratios
• 2 Introduction to Quantum Mechanics
  • 2.1 Laws of Quantum Mechanics
  • 2.2 States, observables and eigenvalues
    • 2.2.1 Properties of eigenfunctions
    • 2.2.2 Review of linear Algebra
  • 2.3 Measurement and probability
    • 2.3.1 Wavefunction collapse
    • 2.3.2 Position measurement
    • 2.3.3 Momentum measurement
    • 2.3.4 Expectation values
  • 2.4 Energy eigenvalue problem
    • 2.4.1 Free particle
  • 2.5 Operators, Commutators and Uncertainty Principle
    • 2.5.1 Commutator
    • 2.5.2 Commuting observables
    • 2.5.3 Uncertainty principle
• 3 Scattering, Tunneling and Alpha Decay
  • 3.1 Review: Energy eigenvalue problem
  • 3.2 Unbound Problems in Quantum Mechanics
    • 3.2.1 Infinite barrier
    • 3.2.2 Finite barrier
  • 3.3 Alpha decay
    • 3.3.1 Energetics
    • 3.3.2 Quantum mechanics description of alpha decay
• 4 Energy Levels
  • 4.1 Bound problems
    • 4.1.1 Energy in Square infinite well (particle in a box)
    • 4.1.2 Finite square well
  • 4.2 Quantum Mechanics in 3D: Angular momentum
    • 4.2.1 Schr¨odinger equation in spherical coordinates
    • 4.2.2 Angular momentum operator
    • 4.2.3 Spin angular momentum
    • 4.2.4 Addition of angular momentum
  • 4.3 Solutions to the Schr¨odinger equation in 3D
    • 4.3.1 The Hydrogen atom
    • 4.3.2 Atomic periodic structure
    • 4.3.3 The Harmonic Oscillator Potential
  • 4.4 Identical particles
    • 4.4.1 Bosons, fermions
    • 4.4.2 Exchange operator
    • 4.4.3 Pauli exclusion principle
• 5 Nuclear Structure
  • 5.1 Characteristics of the nuclear force
  • 5.2 The Deuteron
    • 5.2.1 Reduced Hamiltonian in the center-of-mass frame
    • 5.2.2 Ground state
    • 5.2.3 Deuteron excited state
    • 5.2.4 Spin dependence of nuclear force
  • 5.3 Nuclear models
    • 5.3.1 Shell structure
    • 5.3.2 Nucleons Hamiltonian
    • 5.3.3 Spin orbit interaction
    • 5.3.4 Spin pairing and valence nucleons
• 6 Time Evolution in Quantum Mechanics
  • 6.1 Time-dependent Schr¨odinger equation
    • 6.1.1 Solutions to the Schr¨odinger equation
    • 6.1.2 Unitary Evolution
  • 6.2 Evolution of wave-packets
  • 6.3 Evolution of operators and expectation values
    • 6.3.1 Heisenberg Equation
    • 6.3.2 Ehrenfest’s theorem
  • 6.4 Fermi’s Golden Rule
• 7 Radioactive decay
  • 7.1 Gamma decay
    • 7.1.1 Classical theory of radiation
    • 7.1.2 Quantum mechanical theory
    • 7.1.3 Extension to Multipoles
    • 7.1.4 Selection Rules
  • 7.2 Beta decay
    • 7.2.1 Reactions and phenomenology
    • 7.2.2 Conservation laws
    • 7.2.3 Fermi’s Theory of Beta Decay
• 8 Applications of Nuclear Science
  • 8.1 Interaction of radiation with matter
    • 8.1.1 Cross Section
    • 8.1.2 Neutron Scattering and Absorption
    • 8.1.3 Charged particle interaction
    • 8.1.4 Electromagnetic radiation

1. Introduction to Nuclear Physics

1.1 Basic Concepts 1.1.1 Terminology 1.1.2 Units, dimensions and physical constants 1.1.3 Nuclear Radius 1.2 Binding energy and Semi-empirical mass formula 1.2.1 Binding energy 1.2.2 Semi-empirical mass formula 1.2.3 Line of Stability in the Chart of nuclides 1.3 Radioactive decay 1.3.1 Alpha decay 1.3.2 Beta decay 1.3.3 Gamma decay 1.3.4 Spontaneous fission 1.3.5 Branching Ratios

1.1 Basic Concepts

In this chapter we review some notations and basic concepts in Nuclear Physics. The chapter is meant to setup a common language for the rest of the material we will cover as well as rising questions that we will answer later on.

1.1.1 Terminology

A given atom is specified by the number of

• neutrons: N
• protons: Z
• electrons: there are Z electron in neutral atoms

Atoms of the same element have same atomic number Z. They are not all equal, however. Isotopes of the same element have different # of neutrons N. Isotopes are denoted by AXN or more often by Z

A Z X

where X is the chemical symbol and A = Z + N is the mass number. E.g.: 235 92 U,

(^238) U [the Z number is redundant,

thus it is often omitted]. When talking of different nuclei we can refer to them as

• Nuclide: atom/nucleus with a specific N and Z.
• Isobar: nuclides with same mass # A (�= Z, N ).
• Isotone: nuclides with same N, �= Z.
• Isomer: same nuclide (but different energy state).

We finally obtain the expression for the nuclear binding energy :

Fig. 1: Binding energy per nucleon (B/A in MeV vs. A) of stables nuclides (Red) and unstable nuclides (Gray).

Quantities of interest are also the neutron and proton separation energies:

Z XN^ )^ −^ B( Sn = B(A^ A−^1 XN − 1 ) Z Sp = B(Z^ AXN ) − B(AZ−−^11 XN )

which are the analogous of the ionization energies in atomic physics, reflecting the energies of the valence nucleons. We will see that these energies show signatures of the shell structure of nuclei.

1.2.2 Semi-empirical mass formula

The binding energy is usually plotted as B/A or binding energy per nucleon. This illustrates that the binding energy is overall simply proportional to A, since B/A is mostly constant. There are however corrections to this trend. The dependence of B/A on A (and Z) is captured by the semi-empirical mass formula. This formula is based on first principle considerations (a model for the nuclear force) and on experi mental evidence to find the exact parameters defining it. In this model, the so-called liquid-drop model, all nucleons are uniformly distributed inside a nucleus and are bound together by the nuclear force while the Coulomb interaction causes repulsion among protons. Characteristics of the nuclear force (its short range) and of the Coulomb interaction explain part of the semi-empirical mass formula. However, other (smaller) corrections have been introduced to take into account variations in the binding energy that emerge because of its quantum-mechanical nature (and that give rise to the nuclear shell model). The semi-empirical mass formula (SEMF) is

M (Z, A) = Zm(^1 H) + N mn − B(Z, A)/c^2

where the binding energy B(Z, A) is given by the following formula:

(A − 2 Z)^2

B(A, Z) = av A − asA^2 /^3 − acZ(Z − 1)A−^1 /^3 − asym + δap A−^3 /^4 A

ր ↑ volume surface

We will now study each term in the SEMF.

Coulomb

symmetry

տ pairing

B =

Zmp + N mn − [mA(AX) − Zme]

c^2

A. Volume term

The first term is the volume term av A that describes how the binding energy is mostly proportional to A. Why is that so? Remember that the binding energy is a measure of the interaction among nucleons. Since nucleons are closely packed in the nucleus and the nuclear force has a very short range, each nucleon ends up interacting only with a few neighbors. This means that independently of the total number of nucleons, each one of them contribute in the same way. Thus the force is not proportional to A(A − 1)/ 2 ∼ A^2 (the total # of nucleons one nucleon can interact with) but it’s simply proportional to A. The constant of proportionality is a fitting parameter that is found experimentally to be av = 15.5MeV. This value is smaller than the binding energy of the nucleons to their neighbors as determined by the strength of the nuclear (strong) interaction. It is found (and we will study more later) that the energy binding one nucleon to the other nucleons is on the order of 50 MeV. The total binding energy is instead the difference between the interaction of a nucleon to its neighbor and the kinetic energy of the nucleon itself. As for electrons in an atom, the nucleons are fermions, thus they cannot all be in the same state with zero kinetic energy, but they will fill up all the kinetic energy levels according to Pauli’s exclusion principle. This model, which takes into account the nuclear binding energy and the kinetic energy due to the filling of shells, indeed gives an accurate estimate for av.

B. Surface term

The surface term, −asA^2 /^3 , also based on the strong force, is a correction to the volume term. We explained the volume term as arising from the fact that each nucleon interacts with a constant number of nucleons, independent of A. While this is valid for nucleons deep within the nucleus, those nucleons on the surface of the nucleus have fewer nearest neighbors. This term is similar to surface forces that arise for example in droplets of liquids, a mechanism that creates surface tension in liquids. Since the volume force is proportional to BV ∝ A, we expect a surface force to be ∼ (BV )^2 /^3 (since the surface S ∼ V 2 /^3 ). Also the term must be subtracted from the volume term and we expect the coefficient as to have a similar order of magnitude as av. In fact as = 13 − 18MeV.

C. Coulomb term

The third term −acZ(Z −1)A−^1 /^3 derives from the Coulomb interaction among protons, and of course is proportional to Z. This term is subtracted from the volume term since the Coulomb repulsion makes a nucleus containing many protons less favorable (more energetic). To motivate the form of the term and estimate the coefficient ac, the nucleus is modeled as a uniformly charged sphere. The potential energy of such a charge distribution is

1 3 Q^2 E = 4 πǫ 0 5 R

since from the uniform distribution inside the sphere we have the charge q(r) = 4 πr^3 ρ = Q (^ r^ d^3 and the potential 3 R energy is then:

1 q �r) 1 q(�r) 1 R q(r) E = dq(�r) = d^3 �r ρ = dr πr^2 ρ 4 0 |�r| 4 πǫ 0 |�r| 4 πǫ 0 r

1 R^3 Q r �^3 1 1 R^ Q^2 r 4 1 3 Q^2 = 4 π dr r^2 Q = dr = 4 πǫ 0 0 4 πR^3 R r 4 πǫ 0 0 R^64 πǫ 0 5 R

Using the empirical radius formula R = R 0 A^1 /^3 and the total charge Q^2 = e^2 Z(Z − 1) (reflecting the fact that this term will appear only if Z > 1, i.e. if there are at least two protons) we have :

Q^2 e^2 Z(Z − 1)

R R 0 A^1 /^3

2 which gives the shape of the Coulomb term. Then the constant ac can be estimated from ac ≈ (^3) 5 4^ πǫe 0 R 0 , with R 0 = 1.25fm, to be ac ≈ 0 .691 MeV, not far from the experimental value.

πǫ

ǫ

0

ǫ

0

0

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Fig. 3: Chart of nuclides from http://www.nndc.bnl.gov/chart/. Each nuclide is color-labeled by its half-life (black for stable nuclides)

Fig. 4: Nuclide chart (obtained with the software Mathematica). Left: Z vs. A, Right: Z/A vs. A. In red, stable nuclides. The black line represents Z = A/2.

1.3 Radioactive decay

Radioactive decay is the process in which an unstable nucleus spontaneously loses energy by emitting ionizing particles and radiation. This decay, or loss of energy, results in an atom of one type, called the parent nuclide, transforming to an atom of a different type, named the daughter nuclide. The three principal modes of decay are called the alpha, beta and gamma decays. We will study their differences and exact mechanisms later in the class. However these decay modes share some common feature that we describe now. What these radioactive decays describe are fundamentally quantum processes, i.e. transitions among two quantum states. Thus, the radioactive decay is statistical in nature, and we can only describe the evolution of the expectation values of quantities of interest, for example the number of atoms that decay per unit time. If we observe a single unstable nucleus, we cannot know a priori when it will decay to its daughter nuclide. The time at which the decay happens is random, thus at each instant we can have the parent nuclide with some probability p and the daughter with probability 1 − p. This stochastic process can only be described in terms of the quantum mechanical evolution of the nucleus. However, if we look at an ensemble of nuclei, we can predict at each instant the average number of parent an daughter nuclides. If we call the number of radioactive nuclei N , the number of decaying atoms per unit time is dN/dt. It is found that this rate is constant in time and it is proportional to the number of nuclei themselves:

d N = −λN (t) d t

The constant of proportionality λ is called the decay constant. We can also rewrite the above equation as

dN/dt λ = − N

where the RHS is the probability per unit time for one atom to decay. The fact that this probability is a constant is a characteristic of all radioactive decay. It also leads to the exponential law of radioactive decay :

N (t) = N (0)e^ −λt

We can also define the mean lifetime

τ = 1/λ

and the half-life

t 1 / 2 = ln (2)/λ

which is the time it takes for half of the atoms to decay, and the activity

A(t) = λN (t)

Since A can also be obtained as , the activity can be estimated from the number of decays ΔN during a small time δt such that δt ≪ t 1 / 2. A common situation occurs when the daughter nuclide is also radioactive. Then we have a chain of radioactive decays, each governed by their decay laws. For example, in a chain N 1 → N 2 → N 3 , the decay of N 1 and N 2 is given by:

dN 1 = −λ 1 N 1 dt, dN 2 = +λ 1 N 1 dt − λ 2 N 2 dt

Another common characteristic of radioactive decays is that they are a way for unstable nuclei to reach a more energetically favorable (hence stable) configuration. In α and β decays, a nucleus emits a α or β particle, trying to approach the most stable nuclide, while in the γ decay an excited state decays toward the ground state without changing nuclear species.

1.3.1 Alpha decay

If we go back to the binding energy per mass number plot (B/A vs. A) we see that there is a bump (a peak) for A ∼ 60 − 100. This means that there is a corresponding minimum (or energy optimum) around these numbers. Then the heavier nuclei will want to decay toward this lighter nuclides, by shedding some protons and neutrons. More specifically, the decrease in binding energy at high A is due to Coulomb repulsion. Coulomb repulsion grows in fact as Z^2 , much faster than the nuclear force which is ∝ A.

∣d N∣ d t

Then, the Coulomb term, although small, makes Q increase at large A. We find that Q ≥ 0 for A � 150, and it is Q ≈ 6MeV for A = 200. Although Q > 0, we find experimentally that α decay only arise for A ≥ 200. Further, take for example Francium-200 (^20087 Fr 113 ). If we calculate Qα from the experimentally found mass differences we obtain Qα ≈ 7 .6MeV (the product is 196 At). We can do the same calculation for the hypothetical decay into a (^12) C and remaining fragment ( (^188) Tl 107 ): 81

Z XN^ )^ −^ m(

Q 12 A−^12 ′

C =^ c

(^2) [m(A (^) X N − 6 )^ −^ m(

(^12) C)] ≈ 28 M eV Z− 6

Thus this second reaction seems to be more energetic, hence more favorable than the alpha-decay, yet it does not occur (some decays involving C-12 have been observed, but their branching ratios are much smaller). Thus, looking only at the energetic of the decay does not explain some questions that surround the alpha decay:

• Why there’s no 12 C-decay? (or to some of this tightly bound nuclides, e.g O-16 etc.)
• Why there’s no spontaneous fission into equal daughters?
• Why there’s alpha decay only for A ≥ 200?
• What is the explanation of Geiger-Nuttall rule? log t 1 / 2 ∝ √^1 Qα

1.3.2 Beta decay

The beta decay is a radioactive decay in which a proton in a nucleus is converted into a neutron (or vice-versa). Thus A is constant, but Z and N change by 1. In the process the nucleus emits a beta particle (either an electron or a positron) and quasi-massless particle, the neutrino.

Courtesy of Thomas Jefferson National Accelerator Facility - Office of Science Education. Used with permission. Fig. 6: Beta decay schematics

There are 3 types of beta decay: A A ′ − Z XN^ →^ Z+1XN − 1 +^ e^ +^ ν¯

This is the β−^ decay (or negative beta decay). The underlying reaction is:

n → p + e^ − + ν¯

that corresponds to the conversion of a proton into a neutron with the emission of an electron and an anti-neutrino. There are two other types of reactions, the β+^ reaction,

A (^) → A ′ (^) e+ + ν ⇐⇒ p → n + e+ + ν Z XN^ Z− 1 XN +1 +

which sees the emission of a positron (the electron anti-particle) and a neutrino; and the electron capture:

A − A ′ − Z XN^ +^ e^ →^ Z− 1 XN +1 +^ ν ⇐⇒^ p^ +^ e^ →^ n^ +^ ν

a process that competes with, or substitutes, the positron emission.

Recall the mass of nuclide as given by the semi-empirical mass formula. If we keep A fixed, the SEMF gives the binding energy as a function of Z. The only term that depends explicitly on Z is the Coulomb term. By inspection we see that B ∝ Z^2. Then from the SEMF we have that the masses of possible nuclides with the same mass number lie on a parabola. Nuclides lower in the parabola have smaller M and are thus more stable. In order to reach that minimum, unstable nuclides undergo a decay process to transform excess protons in neutrons (and vice-versa).

(^49) In

(^50) Sn (^51) Sb

(^52) Te

(^53) I

(^55) Cs

(^54) Xe

(^56) Ba

A= (^49) In

(^50) Sn 51 Sb

(^52) Te

(^53) I

(^55) Cs

(^54) Xe

(^56) Ba

A=

(^57) La

Fig. 7: Nuclear Mass Chain for A=125, (left) and A=128 (right)

The beta decay is the radioactive decay process that can convert protons into neutrons (and vice-versa). We will study more in depth this mechanism, but here we want simply to point out how this process can be energetically favorable, and thus we can predict which transitions are likely to occur, based only on the SEMF. For example, for A = 125 if Z < 52 we have a favorable n → p conversion (beta decay) while for Z > 52 we have p → n (or positron beta decay), so that the stable nuclide is Z = 52 (tellurium).

A. Conservation laws

As the neutrino is hard to detect, initially the beta decay seemed to violate energy conservation. Introducing an extra particle in the process allows one to respect conservation of energy. The Q value of a beta decay is given by the usual formula:

′ (^) ) − m e

Q^2

β−^ = [mN (

AX) − m N (

A (^) ]c. Z+1X

Using the atomic masses and neglecting the electron’s binding energies as usual we have

Q^2 β−^ =^ {[mA(

AX) − Zme] − [mA(A ′ (^) ) − (Z + 1)me] − me}c = [mA(AX) − mA(A ′ (^) )]c. Z+1X^ Z+1X

The kinetic energy (equal to the Q) is shared by the neutrino and the electron (we neglect any recoil of the massive nucleus). Then, the emerging electron (remember, the only particle that we can really observe) does not have a fixed energy, as it was for example for the gamma photon. But it will exhibit a spectrum of energy (or the number of electron at a given energy) as well as a distribution of momenta. We will see how we can reproduce these plots by analyzing the QM theory of beta decay. Examples

64 ր^64 − 30 Zn +^ e^ +^ ν,¯^ Qβ^ =^0.^57 M eV 29 Cu^ ց^64 + 28 Ni +^ e^ +^ ν,^ Qβ^ = 0.^66 M eV

The neutrino and beta particle (β±) share the energy. Since © Neil Spooner. All rights reserved. This content is excluded the neutrinos are very difficult to detect (as we will see they from our Creative Commons license. For more information, are almost massless and interact very weakly with matter), the

see http://ocw.mit.edu/fairuse.

electrons/positrons are the particles detected in beta-decay Fig. 8: Beta decay spectra: Distribution of momentum and they present a characteristic energy spectrum (see Fig.

(top plots) and kinetic energy (bottom) for. β−^ (left) and

8).

β+^ (right) decay.

The difference between the spectrum of the β±^ particles is due to the Coulomb repulsion or attraction from the nucleus.

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22.02 Introduction to Applied Nuclear Physics

Spring 2012

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